# Cometrics on Sub-Riemannian manifolds

1. ### Kreizhn

743
Hi all,

This may seem like a simple problem, but I just want to clarify something.

The issue is the relationship between sub-Riemannian manifolds and cometrics. In particular, say we have a manifold M and a cometric on the cotangent bundle T*M. Firstly, it is my understanding that somehow a cometric "induces" a sub-Riemannian manifold; I think this is done by defining the Hamiltonian
$$H(p,q) = \frac12 \langle p,p\rangle, (p,q) \in T_q^*M$$
and using the fact that all sub-Riemannian manifolds are completely characterized by their Hamiltonian. Is this correct?

Secondly, can one go in the other direction. Namely, if one has a sub-Riemannian manifold, is there a naturally defined cometric? If so, can it be defined on the entire cotangent bundle or just some sort of dual to the horizontal distribution?

Naturally above, I'm considering the case when the distribution is not the entire tangent bundle, and when the cometric is degenerate. Otherwise it would seem obvious that the musical isomorphisms would give the necessary relations.

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